Messi, Cristiano Ronaldo, Mbappé, Modric, Luis Suárez, Benzema, Neymar or Lewandowski. The list of emblematic players that appear in the sticker album of this World Cup is one of the most dazzling in history. The presence of so many stars has attracted millions of collectors from all corners of the planet, even becoming a matter of state. In Argentina, the madness unleashed to get Messi’s sticker has been such that it has led to a shortage, and the Government has had to mediate between the manufacturing company and the sellers to stop the shortage and speculation.
While, with good luck, getting the Messi sticker could cost €1 (the price of a pack), speculation has been rampant: for the most unlucky, its price reaches €90.
The goal is to get 670 different stickers to complete the collection. Taking into account that each envelope contains 5 stickers, the minimum price to complete it is €134 if the impossible happens that the collector gets different stickers in all the envelopes. However, without the black market, trading or speculation involved, our calculations conclude that, realistically, one would have to invest €940 with a standard deviation of €170 to get the complete collection of World Cup stickers. The album will come out for about the same price as a simple electric bike. Let us see how we arrive at this result.
A globalist experiment
At the beginning of this type of collection, almost all the stickers that we find in each envelope are new. However, very soon repeated cards begin to appear, the reps. Where any collector would find frustration, we scientists see an opportunity to do an experiment. In this way, I added paper and pencil to my collection of the World Cup, noting how many new stickers I have obtained for each envelope purchased. Although this measure is interesting, I was missing something else to know if my luck has really been good or not. To do this, I decided to create a mathematical model of what was happening.
5,000 families inside a computer
Mathematical models are very useful for scientists because, in reality, many times we only have an observation of the facts. However, considering different random conditions, models can provide us with many observations, increasing the information available.
For example, the curves that described the number of people infected with covid-19 as a function of time were based on a single observation. However, those curves included the consequences of many random causes. To better understand the situation and decide what measures to take, the mathematical models provided many curves of the same phenomenon and even reported the probability of the different scenarios for the evolution of the pandemic.
The model of this collection is very simple. Our computer simulates 5,000 families buying World Cup stickers. Each family will find five random stickers in each package and will continue buying packages until the collection is finished.
The only condition is that there cannot be two identical cards in the same package. We obtain 5,000 curves (one for each family) that represent the number of new trading cards as a function of the number of packages purchased. With all these curves we can obtain the trajectory of the median (curve that at each point has half of the families with the best luck above and the other half below) and the area between the 5th and 95th percentiles.
The area between those percentiles gives us a lot of information, since we need very good luck to be above it (5% better) or very bad to be the opposite. This means that the vast majority of trajectories will be included in this area. As we can see in the figure, my collection has always been within the area between the 5th and 95th percentiles. Also, I seem to have been lucky, because the deviations from the median have been above it.
Changing ‘repes’ stickers can save 300 euros
The model shows us that my experience has been good luck. However, this model can tell us more, such as how many we would need to complete the collection.
By averaging the number of envelopes that each family had to buy to get the 670 stickers, we obtain that €940 would have to be invested with a standard deviation of €170. This result indicates the importance of luck in completing the collection, since it can change the final price by hundreds of euros. By the way, similar models are used in ecology to create rarefaction curves, which show the number of species observed as a function of the number of samples. Under the eye of a statistician, a biologist looking for different species in an ecosystem is not that different from a collector hunting for new cards.
Finally, we have modified our model to include the exchange of ‘repes’. Now our families are grouped in pairs that exchange cards. The exchange is fair, that is, your ‘repe’ chrome in exchange for mine. The result is that the average price to complete the collection is €660 with a standard deviation of €150, a much lower price than in the case of an isolated family. In short, it is better to have a friend than to have good luck.